the symplectic form
/ sim-PLEK-tik /
The symplectic form is the hidden geometry of phase space -- the structure that makes it more than just a 2n-dimensional set of coordinates. Ordinary space has a metric, which measures lengths and angles. Phase space instead has a symplectic form, which measures oriented areas. It is this area-measuring structure, not any notion of distance, that Hamiltonian mechanics respects, and it is the reason positions and momenta pair up the way they do.
On phase space with canonical coordinates (q_i, p_i), the symplectic form is the antisymmetric object omega = sum_i dq_i ^ dp_i (a differential 2-form; the wedge ^ denotes an oriented, antisymmetric product). It assigns to any two-dimensional patch a signed area, summed over the conjugate planes. Its defining features are that it is antisymmetric and non-degenerate (no nonzero direction has zero area-pairing with everything), and closed (d omega = 0). The Hamiltonian generates dynamics through it: the equations of motion are the statement that the vector field of the flow, contracted with omega, equals dH. Canonical transformations are exactly the maps that preserve omega, and the Poisson bracket is omega evaluated on the gradients of two functions.
The symplectic form is the coordinate-free heart of the whole subject. Liouville's theorem is the statement that omega^n (its n-fold wedge, the phase-space volume) is invariant; Poincare's stronger invariants say each power of omega is separately conserved. This geometry also disciplines numerical work: symplectic integrators are designed to preserve omega exactly, which is why they track energy faithfully over astronomically long simulations where ordinary methods drift. Seeing mechanics as symplectic geometry is the modern, coordinate-independent viewpoint.
In one degree of freedom omega = dq ^ dp is just the ordinary oriented area element in the phase plane. The statement that a transformation (q, p) -> (Q, P) is canonical becomes the statement that it preserves area: dQ ^ dP = dq ^ dp. This is why canonical maps in a 2D phase space are exactly the area-preserving maps.
In a two-dimensional phase space the symplectic form is oriented area, and canonical transformations are precisely the area-preserving maps.
A symplectic form is not a metric. It is antisymmetric, so omega(v, v) = 0 for every vector -- there is no notion of the 'length' of a phase-space direction, only of areas spanned by pairs. Confusing the two leads to nonsense like asking for the 'distance' between a position and a momentum.